L. Goldberg and A.S. Fraenkel, Rulesets for Beatty games, Preprint.

A.S. Fraenkel and U. Larsson, Take-away games on Beatty's Theorem and the notion of k-invariance, Preprint.

L. Goldberg and A.S. Fraenkel, Extensions of Wythoff's game, Preprint.

A.S. Fraenkel, Complementary balanced words over an arbitrary finite alphabet.

G.E. Andrews, A.S. Fraenkel and J.A. Sellers,
Characterizing the number of $m$-ary
partitions modulo $m$, *Amer. Math. Monthly*, 122 (2015), no. 9,
880-885, 2015. Copyright November 2015 Mathematical Association
of America. All Rights Reserved.

A.S. Fraenkel and N.B. Ho,
When are translations of *P*-positions of Wythoff's
game *P*-positions?, *INTEGERS* 15A, #A4, 12 pp., Proc. INTEGERS
2013 Conference, The Erdos Centennial Conference, 2015.

F. Chung, C. Greene, J. Hutchinson -- Eds., Contributors; D.E. Knuth,
A.S. Fraenkel, D. Zeilberger, M. Petkovsek, N. Calkin,
R. Canfield, A. Nijenhuis, C.D. Savage, R.P. Stanley, F. Lazbenik --
Contributers,

Herbert S. Wilf (1931-2012) Remembrance,
Notices AMS. vol. 62, No. 4, 346--360, April 2015. Full article at:
http://www.math.ucsd.edu/~fan/wilf/hw-ref.pdf .

A.S. Fraenkel and U. Peled, Harnessing the unwieldy
MEX function, In: *Games of No Chance 4*, Proc. BIRS Workshop on
Combinatorial Games, 2008, Banff, Alberta, Canada, MSRI Publ. (R.J.
Nowakowski, ed.), Vol. 63, Cambridge University Press, Cambridge, pp. 77--94,
2015.

A.S. Fraenkel, The Rat game and the Mouse game, In: *Games of
No Chance 4*, Proc. BIRS Workshop on
Combinatorial Games, 2008, Banff, Alberta, Canada, MSRI Publ. (R.J.
Nowakowski, ed.), Vol. 63, Cambridge University Press, Cambridge, pp. 95--113,
2015.

A.S. Fraenkel, R.A. Hearn and A.N. Siegel, Theory of combinatorial games, Chapter 15 in: Handbook of Game Theory, Volume 4 (H. Peyton Young and Shmuel Zamir, eds.), pp. 811- 859, Elsevier 2015.

A.S. Fraenkel, Integrating and Rejuvenating Old and New Gems,
In *Language, Culture, Computation*, Essayss Dedicated to Yaacov
Choueka on the Occasion of His 75th Birthday, Part I, pp. 37--49, Springer,
2014.

A. Fink, A.S. Fraenkel and C. Santos,
Lim Is not Slim, * Internat. J. Game Theory *
(2014) 43:269-281 DOI 10.1007/s00182-013-0380-z
http://link.springer.com/article/10.1007%2Fs00182-013-0380-z#page-1.

A.S. Fraenkel,
Beating Your Fractional Beatty Game Opponent and: What's
the Question to Your Answer? , in: *Advances in Combinatorics,
in Memory of Herbert S. Wilf, Proc. Waterloo Workshop
on Computer Algebra, * I.S. Kotsireas and E.V. Zima, eds., May 2011, pp.
175-186, Springer 2013.

A.S. Fraenkel and Y. Tanny,
A Class
of Wythoff-Like Games , Proc. INTEGERS Conference, Carrollton, Georgia,
Oct. 26-29, 2011, in: INTEGERS, Electr. J. Combinat. Number Theory
12(B) (2013) #A7, 18pp. Also in *Combinatorial Number Theory*,
pp. 91-107, de Gruyter 2013. [Archive of INTEGERS.]

A.S. Fraenkel,
Ratwyt, In memory of Martin Gardner, *The College Math.
J.*, March 2012, pp. 160-164.

A.S. Fraenkel, The vile, dopey, evil and odious game players,
* Discrete
Math.* 312 (2012) 42-46, special volume in honor of the 80th birthday of Gert Sabidussi.

A.S. Fraenkel, Aperiodic subtraction games, * Electronic J. Combinatorics* vol. 18(2) P19 12pp., 2011.
http://www.combinatorics.org/Journal/journalhome.html

S. Blanco and A.S. Fraenkel, Tromping Games: Tiling with Trominoes, *Proc. INTEGERS Conference*, in honor of the birthdays of Melvyn Nathanson and Carl Pomerance, Carrollton, Georgia, Oct. 14-17, 2009, in: *INTEGERS, Electr. J.
Combinat. Number Theory* 11(A) (2011) #6, 29pp.

U. Larsson, P. Hegarty and A.S. Fraenkel, Invariant and dual subtraction games resolving the Duchene-Rigo conjecture, *Theoretical Computer Science* 412 (2011), 729-735.

A.S. Fraenkel, From enmity to amity. *Amer. Math. Monthly* 117 (2010), 646--648.

A.S. Fraenkel, Complementary Iterated Floor Words and the Flora Game. *SIAM J. on Discrete Math* 24 (2010), 570--588.

E. Duchene, A.S. Fraenkel, R.J. Nowakowski and M. Rigo, Extensions and
restrictions of Wythoff's game preserving its P-positions.
*J. Combinat. Theory A* 117 (2010), 545--567.

J. Bell, E. Charlier, A.S. Fraenkel, M. Rigo, A decision problem for ultimately periodic sets in non-standard numeration systems. *Internat. J. Algebra Computat* 19 (2009), 6, 809--839.

A.S. Fraenkel and E. Reisner,
The game of End-Wythoff, in *Games of No Chance 3*, Proc.
BIRS Workshop on Combinatorial Games, July 2005, Banff, Alberta, Canada,
MSRI Publ. (M.H. Albert and R.J. Nowakowski, eds.), Vol. 56, Cambridge
University Press, Cambridge, pp. 329-347, 2009.

A.S. Fraenkel,
The cyclic Butler University game, in: *Mathematical Wizardry for a Gardner* (E. Pegg Jr, A.H. Schoen, and T. Rodgers, eds.), in honor of Martin Gardner, pp. 97-105, A K Peters, Natick, MA, 2009.

E. Duchene, A.S. Fraenkel, S. Gravier and R.J. Nowakowski,
Another bridge between Nim and Wythoff, *Australasian J. of Combinatorics* 44 (2009), 43--56.

S. Artstein-Avidan, A.S. Fraenkel and V. Sos,
A two-parameter family of an extension of Beatty sequences,
*Discrete Math* 308 (2008) 4578-4588.

A.S. Fraenkel,
Games played by Boole and Galois, * Discrete Appl.
Math.*, 156 (2008) 420-427.

P.J. Davis and A.S. Fraenkel,
Remembering Philip Rabinowitz, *Notices AMS* 54 (2007)
1502-1506. Reprinted, with the permission of the AMS, in *The Birth
of Numerical Analysis* (A. Bultheel, R. Cools, eds.)
World Scientific (2009) 179-185.

A.S. Fraenkel,
The Raleigh game. in: *Combinatorial Number Theory*, Proc. INTEGERS
Conference 2005, in celebration of the 70th birthday of Ronald Graham,
Carrollton, Georgia, Oct. 27-30, 2005; B. Landman, M. Nathanson, J.
Nesetril, R. Nowakowski, C. Pomerance, eds., de Gruyter,
pp. 199-208, 2007. Also appeared in *INTEGERS, Electr. J.
Combinat. Number Theory* 7(2) (2007) #13, 11pp., special
volume in honor of Ron Graham.

A.S. Fraenkel and A. Kontorovich,
The Sierpinski sieve of Nim-varieties and binomial coefficients. in:
*Combinatorial Number Theory*, Proc. Integers Conference
2005, in celebration of the 70th birthday of Ronald Graham, Carrollton,
Georgia, Oct. 27-30, 2005; B. Landman, M. Nathanson, J.
Nesetril, R. Nowakowski, C. Pomerance, eds., de Gruyter,
pp. 209-227, 2007. Also appeared in *INTEGERS, Electr. J.
Combinat. Number Theory* 7(2) (2007) #14, 19pp., special volume in honor
of Ron Graham.

A.S. Fraenkel,
Why are games exciting and stimulating?,
*Math Horizons* February 2007, 5--7; 32--33. A German translation,
by Niek Neuwahl, was
poster-displayed at the ``Games & Science, Science & Games"
exhibition, which took place in Goettingen, July 17 -- Aug 21, 2005.
It was selected as runner up for an EMS Article Competition 2005
for the educated layman.

A.S. Fraenkel,
Nim is easy, chess is hard --- but why??, *J. Internat.
Computer Games Assoc.* 29 (2006) 203--2006. Earlier shorter
version appeared in
*Plus Mag.*
(electronic) issue 40, pluschat section, Sept., 2006.

A.S. Fraenkel,
Euclid and Wythoff games,
*Discrete Math.* 304 (2005) 65--68.

Aviezri S. Fraenkel and Jamie Simpson,
An extension of the periodicity lemma to longer periods,
* Discrete Appl. Math.* 146 (2005) 146-155
(special issue of 12th Annual Symposium on Combinatorial Pattern Matching).
Preliminary version appeared in * Proc. 12th Annual Sympos.
Combinatorial Pattern Matching*, Jerusalem, Israel, July 1--4, 2001
(A. Amir and G. M. Landau, eds.), Lecture Notes in Computer Science,
Springer, Berlin, pp. 98--105) (invited).

A.S. Fraenkel,
New games related to old and new sequences.
INTEGERS, *Electronic J. Combinat. Number Theory* 4
(2004) #G06, 18pp. http://www.integers-ejcnt.org .
First version appeared in *Advances in Computer Games*,
Proc. 10-th Advances in Computer Games Conference (ACG-10),
Nov. 2003, Graz, Austria (H.J. van den Herik, H. Iida, E.A. Heinz,
eds.), pp. 367--382, Kluwer.

A.S. Fraenkel,
Complexity, appeal and challenges of combinatorial games,
Expanded version of a keynote address at Dagstuhl Seminar Feb.~17--22, 2002,
*Theoretical Computer Science* 313 (2004) 393-415, special issue on
Algorithmic Combinatorial Game Theory.

S.P. Fekete, R. Fleischer, A.S. Fraenkel and M. Schmitt,
Traveling salesmen in the presence of competition,
*Theoretical Computer Science* 313 (2004) 377-392, special issue on
Algorithmic Combinatorial Game Theory.

E.D. Demaine, R. Fleischer, A.S. Fraenkel and R.J. Nowakowski,
Open problems at the 2002 Dagstuhl Seminar on Algorithmic
combinatorial game theory,
*Theoretical Computer Science* 313 (2004) 539-543, special issue on
Algorithmic Combinatorial Game Theory (Appendix B).

A.S. Fraenkel and Dalia Krieger,
The structure of complementary sets of integers: a 3-shift theorem,
*Internat. J. Pure and Appl. Math.* 10 (2004) 1--49.

Aviezri S. Fraenkel and Ofer Rahat, Complexity of error-correcting codes derived from combinatorial games, Proc. Intern. Conference on Computers and Games CG'2002, Edmonton, Alberta, Canada, July 2002 (Y. Bj\"{o}rnsson, M.\ M\"{u}ller, J. Schaeffer, eds.), Lecture Notes in Computer Science LNCS 2883, pp. 201--212, Springer 2003.

Aviezri S. Fraenkel, Jamie Simpson and Mike Paterson,
On
abelian circular squares in binary words. In: Paul Erdos and his
Mathematics II, Bolyai Soc., Mathematical Studies 11, Budapest 2002,
pp. 329--338. An earlier version entitled ``On weak circular squares
in binary words'' appeared in * Proc. 8th Annual Symposium on
Combinatorial Pattern Matching*, Aarhus, Denmark, June 30--July 2,
1997 (A. Apostolico and J. Hein, eds.), Lecture Notes in Computer
Science, Springer, Berlin, pp. 76--82).

A.S. Fraenkel,
Two-player games on cellular automata, in * More Games
of No Chance*, Proc. MSRI Workshop on Combinatorial Games, July
10--28, 2000, Berkeley, CA (R. J. Nowakowski, ed.), Cambridge
University Press, Cambridge, Vol. 42, 2002, pp. 279--306.

A.S. Fraenkel,
Arrays, numeration systems and Frankenstein games,
*Theoretical Computer Science* 282 (2002) 271--284 (special "FUN
in Algorithms" issue).

A.S. Fraenkel,
Mathematical Chats Between Two Physicists, in *
Puzzlers' Tribute: A Feast for the Mind*, honoring Martin Gardner
(D. Wolfe and T. Rodgers, eds.), A K Peters, 2002, pp. 315--325.

A.S. Fraenkel, Virus versus mankind, in: Proc. 2nd Intern. Conference on Computer Games CG'2000, Hamamatsu, Japan, Oct. 2000, Lecture Notes in Computer Science, Springer (T. Marsland, I. Frank, eds.), 2001, pp. 204--213.

A.S. Fraenkel, T. Seeman and J. Simpson
The subword complexity of a two-parameter family of sequences,
* Electronic J. Combinatorics* vol. 8(2) R10, 2001
http://www.combinatorics.org/Journal/journalhome.html

A.S. Fraenkel, A Selection Of Mathematical Board Games,
Abstract Presented at * Board Games in Academia* III, Florence,
April 12--15, 1999. Proceedings Board Games in Academia III, pp. 46--67.

A.S. Fraenkel and D. Zusman,
A New Heap Game, * Theoretical Computer Science* 252 (2001)
5--12 (special "Computers and Games issue"). Initial version appeared
in Lecture Notes in Computer Science 1558, Proc. * 1-st Intern.
Conference on Computer Games, Tsukuba, Japan, Nov. 1998*, pp. 205--211,
1999.

A.S. Fraenkel and O. Rahat,
Infinite cyclic impartial games, * Theoretical Computer Science*
252 (2001) 13-22 (special "Computers and Games issue"). Initial version
appeared in Lecture Notes in Computer Science 1558,
Proc. * 1-st Intern. Conference on Computer Games, Tsukuba,
Japan, Nov. 1998*, pp. 212--221, 1999.

A.S. Fraenkel,
Recent results and questions in combinatorial game complexities,
Expanded version of invited lecture at Ninth Australasian Workshop
on Combinatorial Algorithms, Perth, Western Australia, 27---30 July
1998, * Theoret. Comput. Sci.* 249 (2000) 265--288. Conference
version in: Proc. AWOCA98 --- Ninth Australasian Workshop on Combinatorial
Algorithms, (C.S. Iliopoulos, ed.), pp. 124--146.

A.S. Fraenkel,
On the recurrence *f(m+1)=b(m)f(m)-f(m-1)* and applications,
* Discrete Mathematics* 224 (2000) 273--279 (Note).

A.S. Fraenkel and S.T. Klein,
Information
retrieval from annotated texts, * J. Amer. Soc.
for Information Sciences* 50 (1999) 845--854.

A.S. Fraenkel and Jamie Simpson,
The exact number of squares in Fibonacci words,
* Theoret. Comput. Sci.* 218 (1999) 95--106 (special *Words*
issue).

A.S. Fraenkel, Protein folding, spin glass and computational complexity, Expanded form of invited paper presented at DNA Based Computers III, Proc. DIMACS Workshop (H. Rubin and D. H. Wood, Eds.), June 23--25, 1997, University of Pennsylvania, DIMACS Series in Discrete Mathematics and Theoretical Computer Science Vol. 48, AMS, 1999, pp. 101--121.

A.S. Fraenkel,
Heap games, numeration systems and sequences,
*Annals of Combinatorics* 2 (1998) 197--210.

U.Blass, A.S. Fraenkel and R. Guelman,
How far can Nim in disguise be stretched?,
*J. Combin. Theory* (Ser. A) 84 (1998) 145--156.

A.S. Fraenkel,
The Responsa storage and retrieval system---whither?
The Hebrew version:
Proyect hashut: avar, hoveh---v'gam atid?
appeared in * Dinei Yisrael* 19 (1997-98) 253--270.

A.S. Fraenkel and Michal Ozery,
Adjoining to Wythoff's game its P-positions as moves,
* Theoretical Computer Science* 205 (1998) 283--296.

A.S. Fraenkel,
Multivision: a game of arbitrarily long play,
*Amer. Math. Monthly* 105 (1998) 923--928.

A.S. Fraenkel and Jamie Simpson,
How many squares can a string contain?
*J. Combin. Theory Ser. A* 82 (1998) 112--120.
Errata: In Table 1, the number of maximal words (last column)
should be 6 and 72 for n=3 and n=12 respectively. These corrections
do not otherwise affect the results of the paper. We thank Troy
Vasiga for pointing out the first of these typos.

A.S. Fraenkel,
Combinatorial game theory foundations applied to digraph kernels
, *Electronic J. of
Combinatorics* 4(2) (1997) R10, 17pp. (Wilf Festschrift).

2007, Elected *Meritocrat of Rehovot* (Yakir Ha'ir).
One of the photos of the ceremony,
March 6, 2007. (without recipient and spouse)

2007, Israel Prize for the Responsa Project, initiated in 1963, that appears to be the first viable, durable full-text retrieval system, 35 years prior to Google.

2006, Recipient of *WEIZAC Medal*, at December 5 special seminar,
celebrating recognition by the IEEE of the WEIZAC project as a Milestone
in the History of Computers. (WEIZAC, constructed at the Weizmann Institute
of Science in 1954, was the first electronic digital computer in the
Middle East, and one of the first in the world.)

2005, *Euler Medal* recipient of the Institute for Combinatorics and Its
Applications. Excerpts from the citation in
*Bull. ICA* 47, May 2006;
front of medal,
back; photographed by Eli Shechtman.

2005, *Runner up* in European Mathematical Society
Article Competition.

2001, The Fraenkel *Festschrift* volume,
*Electr, J. of Combinatorics*,
vol. 8(2).

1998, *Quality Initiative Citation* to the Responsa Project for
creative, high-quality and visionary projects to celebrate
the 50th anniversary of the establishment of the State of Israel.

1972, Feder Foundation Prize for initiating and creating the Responsa Project.

*Wikipedia* pages: Hebrew
;
English ; German
.

- Email: "aviezri.fraenkel" followed by the "at" symbol, followed by "weizmann.ac.il" .
- Telephone: 972-8-9343539; secretary: 972-8-9343545
- Fax: 972-8-9342945.
- Business address:
- Professor Aviezri S. Fraenkel
- Dept of Compu Sci & Appl Math
- Weizmann Institute of Science
- 76100 Rehovot
- Israel

URL for this homepage is: http://www.wisdom.weizmann.ac.il/~fraenkel

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